Question

Given the following:

f(x)=2x2−x−6


g(x)=x2+3x−10


What is (fg)(x)
? Then, find the domain of (fg)(x)
.

(2 points)

Answer 1

To find \( (fg)(x) \), we need to multiply the two functions \( f(x) \) and \( g(x) \).

Given:
\[
f(x) = 2x^2 - x - 6
\]
\[
g(x) = x^2 + 3x - 10
\]

Calculating \( (fg)(x) \):
\[
(fg)(x) = f(x) \cdot g(x) = (2x^2 - x - 6)(x^2 + 3x - 10)
\]

Now let's perform the multiplication:

Expanding:
\[
(2x^2 - x - 6)(x^2 + 3x - 10) = 2x^2(x^2) + 2x^2(3x) + 2x^2(-10) - x(x^2) - x(3x) - x(-10) - 6(x^2) - 6(3x) - 6(-10)
\]

Calculating term by term:
\[
= 2x^4 + 6x^3 - 20x^2 - x^3 - 3x^2 + 10x - 6x^2 - 18x + 60
\]

Now, combine like terms:
\[
= 2x^4 + (6x^3 - x^3) + (-20x^2 - 3x^2 - 6x^2) + (10x - 18x) + 60
\]
\[
= 2x^4 + 5x^3 - 29x^2 - 8x + 60
\]

Thus,
\[
(fg)(x) = 2x^4 + 5x^3 - 29x^2 - 8x + 60
\]

### Finding the Domain of \( (fg)(x) \)

Both \( f(x) \) and \( g(x) \) are polynomials, and the domain of a polynomial function is always all real numbers. Therefore, the domain of \( (fg)(x) \) is also all real numbers.

**Final Answers:**
- \( (fg)(x) = 2x^4 + 5x^3 - 29x^2 - 8x + 60 \)
- Domain of \( (fg)(x) \): all real numbers, \( (-\infty, \infty) \).